Kernel-based Discretisation for Solving Matrix-Valued PDEs

TL;DR

This paper introduces a novel kernel-based meshfree discretisation method for solving matrix-valued PDEs, with applications in dynamical systems, extending existing kernel theory to matrix-valued functions and providing error estimates.

Contribution

It develops a new tensor-valued kernel approach for matrix PDEs, extending reproducing kernel Hilbert space theory and analyzing error bounds for the discretisation.

Findings

Successful application to a dynamical systems example

Derivation of error estimates for the approximation

Extension of kernel methods to matrix-valued functions

Abstract

In this paper, we discuss the solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyse a new meshfree discretisation scheme using kernel-based approximation spaces. However, since these approximation spaces have now to be matrix-valued, the kernels we need to use are fourth order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with a typical example from dynamical systems.